# 'Shifted' Poisson Processes?

#### TeoNichelli

##### New Member
Hi all.

I’m a graduate student in ops research, but my research is in the optimization side of things more so than the stochastic processes. I need a suggestion about how to solve this inventory-motivated problem, let me explain it briefly.

Suppose products are sold from a warehouse according to a Poisson process with rate lambda. But, an empirically determined percentage of all sales, say pi, will be returned to the warehouse. We’ll assume for simplicity that we can instantly resell the products and that they’re non-perishable (i.e. a customer would view it to be as good as a non-used item).

Here’s the catch: the amount of time it takes for a return to come back is exponentially distributed with parameter beta. So, even if I viewed the potential returns leaving the warehouse as being a Poisson process with rate pi*lambda, which I’ve been trying to do, we still have to account for the fact that their ‘return time’ is also stochastic.

How do I determine the return distribution in the long run? That is, from the warehouse’s perspective, with what sort of interarrival distribution should I expect to see the returns arrive? My gut tells me it’s exponential with parameter pi*lambda, since in the long run, we’re just shifting’ the Poisson process by a mean of 1/beta. But, my gut also tells me that since we’re adding so much variance to the interarrivals, this might not be the case.

Any ideas on what kind of models exist to explain this sort of situation? Again, I apologize for any naivete, but I don’t work in stochastic processes.

#### TeoNichelli

##### New Member
Re: Shifted’ Poisson Processes?

With a few more hours of thought, I decided to ignore the return distribution, since as an inventory-problem, I’m really more concerned about the inventory at any given point in time. So, using a merged Poisson process, I consider the total arrival rate’ of demands and returns, and model a countable-state Markov chain, where returns shift the state by +1 and demands shift the state by -1.

Still, though, even though I dodged the analysis in my original post, I’m still curious about a possible solution, if anyone can point me.

#### BGM

##### TS Contributor
Re: Shifted’ Poisson Processes?

Recently I have answer a similar topic, using a similar technique, so I left the details for you to fill up:

Let

$I_0$ be the initial ($t = 0$) inventory level of the warehouse;

$I(t)$ be the inventory level at time $t$;

$X(t) \sim \text{Poisson}(\lambda t)$ be the the number of products sold before time $t$, which is modeled as a Poisson process with rate $\lambda$;

$W_k$ be the time of sale of the $k$-th product;

$R_k \sim \text{Bernoulli}(\pi)$ be the indicator, which equals to $1$ when the $k$-th product is returned;

$Y_k$ be a sequence of i.i.d. random variables, which is the time needed to return to the warehouse if the $k$-th product is returned;

Then we have

$I(t) = I_0 - X(t) + \sum_{k=1}^{X(t)} R_k\mathbf{1}\{W_k + Y_k \leq t\}$

Here we assume $I_0$ is sufficiently large and $t$ is not too large; otherwise it has to capped at 0 to prevent it goes negative. As long as $\pi < 1$ we know that the warehouse will go empty one day if there is no external replenishment.